4 Theor Supplement Section M 71 ommon face 2 1 Figure M.52: egion formed b pasting together 1 2 includes the integral over the same face, but oriented in the opposite direction. Thus, when we add the integrals together, the contributions from the common face cancel, we get the flu integral through. Thus we have F d F d + F d. 1 2 ut we also have div F d div F d + div F d. 1 2 So the Divergence Theorem for follows from the Divergence Theorem for 1 2. Hence we have proved the Divergence Theorem for an region formed b pasting together regions that can be smoothl parameteried b rectangular solids. Eample 1 Let be a spherical ball of radius 2, centered at the origin, with a concentric ball of radius 1 removed. Using spherical coordinates, show that the proof of the Divergence Theorem we have given applies to. Solution e cut into two hollowed hemispheres like the one shown in Figure M.5,. In spherical coordinates, is the rectangle 1 ρ 2, 0 φ π, 0 θ π. Each face of this rectangle becomes part of the boundar of. The faces ρ 1 ρ 2 become the inner outer hemispherical surfaces that form part of the boundar of. The faces θ 0 θ π become the two halves of the flat part of the boundar of. The faces φ 0 φ π become line segments along the -ais. e can form b pasting together two solid regions like along the flat surfaces where θ constant. θ π φ π θ 0 φ π ρ 1 ρ 2 φ π 2 ρ Figure M.5: The hollow hemisphere the corresponding rectangular region in ρθφ-space 1 π θ

5 72 Theor Supplement Section M Proof of Stokes Theorem onsider an oriented surface, bounded b the curve. e want to prove Stokes Theorem: curl F d F d r. e suppose that has a smooth parameteriation r r (s, t), so that corresponds to a region in the st-plane, corresponds to the boundar of. See Figure M.54. e prove Stokes Theorem for the surface a continuousl differentiable vector field F b epressing the integrals on both sides of the theorem in terms of s t, using Green s Theorem in the st-plane. First, we convert the line integral F d r into a line integral around : F d r F r ds + F r dt. So if we define a 2-dimensional vector field G (G 1, G 2 ) on the st-plane b then G 1 F r F d r G 2 F r, G d s, using s to denote the position vector of a point in the st-plane. hat about the flu integral curl F d that occurs on the other side of Stokes Theorem? In terms of the parameteriation, curl F d In Problem 7 on page 74 we show that Hence e have alread seen that curl F r r ds dt. curl F r r G 2. curl F d F d r ( G2 G ) 1 ds dt. G d s. Green s Theorem, the right-h sides of the last two equations are equal. Hence the left-h sides are equal as well, which is what we had to prove for Stokes Theorem. t s Figure M.54: region in the st-plane the corresponding surface in -space; the curve corresponds to the boundar of

6 Theor Supplement Section M 7 Problems for Section M 1. Let be a solid circular clinder along the -ais, with a smaller concentric clinder removed. Parameterie b a rectangular solid in rθ-space, where r, θ, are clindrical coordinates. 2. In this section we proved the Divergence Theorem using the coordinate definition of divergence. Now we use the Divergence Theorem to show that the coordinate definition is the same as the geometric definition. Suppose F is smooth in a neighborhood of ( 0, 0, 0), let U be the ball of radius with center ( 0, 0, 0). Let m be the minimum value of div F on U let M be the maimum value. (a) Let S be the sphere bounding U. Show that S F d m M. olume of U (b) Eplain wh we can conclude that lim 0 S F d div F olume of U ( 0, 0, 0). (c) Eplain wh the statement in part (b) remains true if we replace U with a cube of side, centered at ( 0, 0, 0). Problems 6 fill in the details of the proof of the Divergence Theorem.. Figure M.51 on page 69 shows the solid region in space parameteried b a rectangular solid in stuspace using the continuousl differentiable change of coordinates r r (s, t, u), a s b, c t d, e u f. ( ) Suppose that r r r is positive. (a) Let 1 be the face of corresponding to the face s a of. Show that r, if it is not ero, points into. (b) Show that r r is an outward pointing normal on 1. (c) Find an outward pointing normal on 2, the face of where s b. 4. Show that for the other five faces of the solid in the proof of the Divergence Theorem (see page 70): i F d S i G d S, i 2,, 4, 5, Suppose that F is a continuousl differentiable vector field that a, b, c are vectors. In this problem we prove the formula grad( F b c ) a + grad( F c a ) b + grad( F a b ) c ( a b c ) div F. (a) Interpreting the divergence as flu densit, eplain wh the formula makes sense. [Hint: onsider the flu out of a small parallelepiped with edges parallel to a, b, c.] (b) Sa how man terms there are in the epansion of the left-h side of the formula in artesian coordinates, without actuall doing the epansion. (c) rite down all the terms on the left-h side that contain F 1/. Show that these terms add up to a b c F1. (d) rite down all the terms that contain F 1/. Show that these add to ero. (e) Eplain how the epressions involving the other seven partial derivatives will work out, how this verifies that the formula holds. 6. Let F be a smooth vector field in -space, let (s, t, u), (s, t, u), (s, t, u) be a smooth change of variables, which we will write in vector form as r r (s, t, u) (s, t, u) i +(s, t, u) j +(s, t, u) k. Define a vector field G (G 1, G 2, G ) on stu-space b G 1 F r r G F r r. (a) Show that + G2 + G G 2 F r r F r r + F r r + F r r. (b) Let r 0 r (s 0, t 0, u 0), let a r ( r 0), r b ( r 0), Use the chain rule to show that ( ) G1 + G2 + G r r 0 r c ( r 0). grad( F b c ) a + grad( F c a ) b + grad( F a b ) c.

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